Sumsets of Pisot and Salem numbers

Let S and T   be the sets of Pisot and Salem numbers, respectively. We prove that the set mT∩TmT∩T is empty for every positive integer m⩾2m⩾2, i.e., that no sum of several Salem numbers is a Salem number. We also obtain a result which implies that the sets mT∩SmT∩S and mS∩TmS∩T are nonempty for every m⩾2m⩾2, i.e., that certain Salem numbers can sum to a Pisot number and that certain Pisot numbers can sum to a Salem number. As an explicit example, the Salem number (22+105+5105+14+505+70)/8=10.99925… is expressed by a sum of two Pisot numbers.